Bosons and Fermions can be either fundamental or composite. The fundamental fermions are leptons and quarks. The fundamental bosons are photons, gluons, weak force carriers (W and Z), and gravitons (theoretical).
Bosons have integer spin, fermions have half-integer spins. For example mesons (quark + anti-quark) are bosons, while baryons (3 quarks) are fermions. Atoms may be bosons or fermions, depending on whether the number of neutrons is even (bosons) or odd (fermions).